iGCSE (2021 Edition)

# 11.11 Optimisation problems

Worksheet
Optimisation
1

A manufacturer of men's shirts determines that her profit P varies with respect to the number of shirts produced according to P = 21 x - 0.2 x^{\frac{3}{2}} - 500.

a

Find the derivative, P'.

b

Find the x-value of the stationary point.

c

Find the second derivative, P''.

d

Is the stationary point a maximum or a minimum?

e

What number of shirts corresponds to a maximum profit?

2

A farmer wants to make a rectangular paddock with an area of 4000\text{ m}^2 . She wants to minimize the perimeter of the paddock to keep fencing costs down. If x is the length of one of the sides, the perimeter is given by the equation P = 2 x + \dfrac{8000}{x}.

a

Find the derivative, P'.

b

Find the exact x-values of the stationary points.

c

Find the second derivative, P''.

d

Is the second derivative at x = 20 \sqrt{10} positive or negative?

e

Is the second derivative at x = - 20 \sqrt{10} positive or negative?

f

If length is always positive, is it reasonable that the stationary point at x = 20 \sqrt{10} is a minimum?

g

Find the dimensions of the rectangular paddock, to one decimal place.

3

A pool is being emptied and the volume of water, V litres, left in the pool after t minutes is given by the equation V = 1500 \left(11 - t\right)^{3}, for 0 \leq t \leq 11.

a

Find V', the rate of change of the volume in terms of t minutes.

b

At what rate is the volume of water in the pool changing after 10 minutes?

c

At what time t, for 0 \leq t \leq 11, will the pool be emptied at the fastest rate?

4

A cylinder has a height of h\text{ cm} and a radius that is 90\text{ cm} less than the height.

a

Write an expression for the radius in terms of h.

b

Write an expression for the volume of the cylinder, V, in terms of h.

c

Find the possible two values of h that maximise the volume of the cylinder.

d

Find the first derivative, V'.

e

Find the second derivative, V''.

f

Find the value of h for which the volume of the cylinder is a maximum.

g

Hence, find the maximum volume in exact form.

h

Find the radius for which the volume of the cylinder is a maximum.

Optimisation involving motion
5

The height in meters of a projectile above flat ground is given by h = 9 + 8 t - t^{2}, where t is given in seconds.

a

What is the initial height of the projectile?

b

Find the derivative h'.

c

Find the t-value of the stationary point.

d

Find the coordinates of the stationary point.

e

Find the second derivative, h''.

f

Is the stationary point a maximum or minimum turning point?

6

A boy stands on the edge of a sea-cliff with a height of 48 m. He throws a stone off the cliff so that its vertical height above the cliff is given by h = 16 t - 4 t^{2} where t is given in seconds.

a

Find the derivative h'.

b

Find the coordinates of the stationary point.

c

Find the second derivative, h''.

d

Is the stationary point \left(2, 16\right) a maximum or minimum turning point?

7

The distance, d, in meters travelled by a train between two stations is given by: d = 0.88 t^{2} - 0.0084 t^{3} where t is measured in seconds.

a

Find the maximum velocity, V, of the train to the nearest m/s.

b

Find the maximum acceleration, A, of the train.

8

The cost, C, in dollars for running a boat at speed, v km/h, is C = \dfrac{v^{2}}{10} - 9 v + 265.

a

Find the derivative, C'.

b

Find the coordinates of the stationary point.

c

Find the second derivative, C''.

d

Is the stationary point a maximum or minimum turning point?

e

Find the minimum running cost.

9

Consider the sum S = \sqrt{a} + \sqrt{b}, where a, b \geq 0 and a + b = 16.

a

Write S in terms of a.

b

Find the value of a at the stationary point of S \left( a \right).

c

Find the second derivative, S'' \left( a \right).

d

Is the stationary point a maximum or minimum?

e

Find the maximum value of S.

10

Charlie is fencing off a rectangular section of his backyard to use for a vegetable garden. He uses the existing back wall and has 24 metres of fencing to create the other three sides. He wishes to make the area for vegetables as large as possible.

a

Write an equation for the length, L, of the vegetable garden in terms of the width, W.

b

Hence write an equation for the area, A, of the vegetable garden in terms of the width, W.

c

Solve for the width, W that could maximise the area of the vegetable garden.

d

Find the second derivative, A''.

e

Find the length of the vegetable garden that maximises the area.

11

A box without cover is to be constructed from a rectangular cardboard that measures 90\text{ cm} by 42\text{ cm} by cutting out four identical square corners of the cardboard and folding up the sides:

Let x be the height of the box, and V the volume of the box.

a

Form an equation for V in terms of x.

b

Find the possible value(s) of x that could correspond to a box with the maximum volume.

c

Find the maximum volume of the box.

12

A rectangular sheet of cardboard measuring 96\text{ cm} by 60\text{ cm} is to be used to make an open box. A square of width w\text{ cm} is removed from each corner to make the net shown.

a

Write an expression for the area, A, of the base of the box in terms of w.

b

Write an expression for the volume, V, of the box in terms of w.

c

Find the possible value(s) of w that could maximise the volume of the box.

d

Find the second derivative, V''.

e

Find the dimensions of the box with the maximum volume.

13

A manufacturer makes cylindrical tins of volume 600\text{ cm}^3. The manufacturer wishes to make tins with the smallest possible surface area that still meet the volume requirement.

a

Write an equation for the height, h, of the cylinder in terms of the radius, r.

b

Hence write an equation for the surface area, A, of the cylinder in terms of the radius, r.

c

Find the radius r that could minimise the surface area.

d

Find the second derivative, A''.

e

Find the minimum surface area.

f

Find the height of the cylinder with the minimum surface area.

14

The organisers of a fundraising event are trying to work out what they should charge per ticket to receive the maximum possible revenue.

They expect 1200 people if they charge \$11 per ticket. For each \$0.25 drop in price of the ticket, they expect an extra 120 people to attend.

a

Write an expression for the price of a ticket after x drops in price of \$0.25 each drop. b Write an expression for the number of people expected to attend after x drops of \$0.25 in the cost of a ticket.

c

Write an equation for the revenue, R, they can expect to receive in terms of x.

d

Find the possible value of x that maximises the revenue.

e

Find the second derivative, R''.

f

Find the price per ticket that will maximise revenue.

g

Find the number of people attending that will maximise revenue.